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 A304865 O.g.f. A(x) satisfies: [x^n] exp( n*(n+4) * x ) / A(x) = 0 for n>0. 3
 1, 5, 37, 434, 7751, 193101, 6200872, 242667316, 11144759839, 585318934391, 34511211188637, 2253285826341378, 161201686356627524, 12530568505972885004, 1051099249634285619168, 94603882448795669308980, 9092091650779263675187695, 929177036869575506758681035, 100608724821944458615599713935, 11504982932704269804549116593702, 1385525417578463389730054278506959 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS EXAMPLE O.g.f.: A(x) = 1 + 5*x + 37*x^2 + 434*x^3 + 7751*x^4 + 193101*x^5 + 6200872*x^6 + 242667316*x^7 + 11144759839*x^8 + 585318934391*x^9 + 34511211188637*x^10 + ... ILLUSTRATION OF DEFINITION. The table of coefficients of x^k/k! in exp(n*(n+4)*x) / A(x) begins: n=0: [1, -5, -24, -1134, -100608, -14542200, -3095496000, -907608905280, ...]; n=1: [1, 0, -49, -1744, -128763, -17383400, -3572628125, -1024052930280, ...]; n=2: [1, 7, 0, -2430, -189600, -22895928, -4410982656, -1218708054720, ...]; n=3: [1, 16, 207, 0, -250107, -33107544, -5910144669, -1540910769048, ...]; n=4: [1, 27, 680, 13970, 0, -42775928, -8486494016, -2090851421760, ...]; n=5: [1, 40, 1551, 56376, 1681797, 0, -10852876125, -2994692165280, ...]; n=6: [1, 55, 2976, 156546, 7748832, 316211400, 0, -3807596825280, ...]; n=7: [1, 72, 5135, 360920, 24718725, 1597879072, 85448027299, 0, ...]; n=8: [1, 91, 8232, 738450, 65376768, 5650680456, 462123838656, 31350065660352, 0, ...]; ... RELATED SERIES. The logarithmic derivative of A(x) yields: A'(x)/A(x) = 5 + 49*x + 872*x^2 + 22661*x^3 + 759915*x^4 + 30843448*x^5 + 1459277062*x^6 + 78529473925*x^7 + 4724556111179*x^8 + 313739794874469*x^9 + ... 1 - 1/A(x) = 5*x + 12*x^2 + 189*x^3 + 4192*x^4 + 121185*x^5 + 4299300*x^6 + 180081132*x^7 + 8675950464*x^8 + 471853727865*x^9 + 28563862383700*x^10 + ... PROG (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m+3) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]} for(n=0, 25, print1( a(n), ", ")) CROSSREFS Cf. A304322, A304318, A304319, A304863, A304864. Sequence in context: A235345 A318002 A323567 * A003709 A286928 A321042 Adjacent sequences:  A304862 A304863 A304864 * A304866 A304867 A304868 KEYWORD nonn AUTHOR Paul D. Hanna, May 19 2018 STATUS approved

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Last modified August 13 05:18 EDT 2022. Contains 356077 sequences. (Running on oeis4.)